A DUALITY PRINCIPLE FOR SELECTION GAMES
LIONEL LEVINE, SCOTT SHEFFIELD AND KATHERINE E. STANGE
Abstract. A dinner table seats k guests and holds n discrete morsels of
food. Guests select morsels in turn until all are consumed. Each guest has a
ranking of the morsels according to how much he would enjoy eating them;
these rankings are commonly known.
A gallant knight always prefers one food division over another if it provides strictly more enjoyable collections of food to one or more other players
(without giving a less enjoyable collection to any other player) even if it
makes his own collection less enjoyable. A boorish lout always selects the
morsel that gives him the most enjoyment on the current turn, regardless
of future consumption by himself and others.
We show the way the food is divided when all guests are gallant knights
is the same as when all guests are boorish louts but turn order is reversed.
This implies and generalizes a classical result of Kohler and Chandrasekaran
(1971) about two players strategically maximizing their own enjoyments.
We also treat the case that the table contains a mixture of boorish louts
and gallant knights.
Our main result can also be formulated in terms of games in which
selections are made by groups. In this formulation, the surprising fact is
that a group can always find a selection that is simultaneously optimal for
each member of the group.
Suppose that k guests are seated at a table containing n discrete morsels
of food (sushi rolls, say). Each player i has a strict ranking <i of the n
morsels according to how enjoyable they are to eat. Players take turns selecting and consuming a morsel from the table, according to a fixed turn order
P1 , P2 , . . . , Pm , where m ≤ n, and Pt ∈ {1, 2, . . . , k} is the player who selects
a morsel on the t th turn.
Our dinner guests do not simply play to maximize their own enjoyment.
Instead, each guest conforms to one of two stereotypes: he is either a gallant
knight or a boorish lout, as explained below.
A plate is a subset of the morsels. Each player’s ranking of the morsels
determines a corresponding partial ordering on plates of a fixed size according
to pairwise comparison: Given two plates A and A0 of the same size, write
A ≤i A0 if there is a bijection f : A → A0 such that a ≤i f (a) for all a ∈ A.
Given a sequence a = (a1 , . . . , am ) of morsel selections representing the play
of a game, let Ai (a) = {at | Pt = i} be the plate eaten by player i. For each
Date: February 13, 2012.
2010 Mathematics Subject Classification. 91A10, 91A18, 91A05, 91A50.
Key words and phrases. agreement, cake cutting, fair division, sequential selection, subgame perfect equilibrium.
The first author was supported by NSF MSPRF 0803064. The second author was
partially supported by NSF grant DMS 0645585. The third author was supported by NSF
MSPRF 0802915.
1
2
LEVINE, SHEFFIELD, STANGE
player we define two partial orders on play sequences, her knight order and
lout order. We define a < b in player i’s knight order if either
(K1) Aj (a) ≤j Aj (b) for all j 6= i and Aj (a) <j Aj (b) for at least one j 6= i;
or
(K2) Aj (a) = Aj (b) for all j 6= i and Ai (a) <i Ai (b).
If neither (K1) nor (K2) holds, and neither holds with the roles of a and b
reversed, and a 6= b, then a and b are incomparable in player i’s knight order.
This could happen for various reasons:
• Some player j 6= i receives incomparable plates Aj (a) and Aj (b); or
• Aj (a) < Aj (b) and Aj 0 (a) > Aj 0 (b) for some j, j 0 6= i; or
• Aj (a) = Aj (b) for all players j.
Player i is a gallant knight if his (partial or total) preference ordering on
play sequences extends his knight order. Thus, a gallant knight must prefer b
to a if all other players receive more enjoyable plates in b than in a. He must
also prefer b to a if all other players receive identical plates in a and b but he
himself receives a more enjoyable plate in b (This can only happen if m < n,
i.e., if not all food is consumed during the game). Our definition only requires
a gallant knight to be “gallant” in a fairly weak sense: he prefers b over a
whenever b is obviously more enjoyable for the other players given their food
preference rankings.
We define a < b in player i’s lout order if the following holds:
(L) as = bs for 1 ≤ s ≤ t − 1 and at <i bt , for some t such that Pt = i.
Player i is a boorish lout if his preference ordering on play sequences extends his
lout order. In other words, a boorish lout cares only about his own enjoyment
and discounts his own future enjoyment so much that he always selects the
morsel that maximizes his enjoyment on the current turn (regardless of the
effect this may have on future turns).
Theorem 1. Under optimal play, the division of the food among gallant
knights playing in turn sequence P1 , . . . , Pm is the same as the division of
the food among boorish louts playing in the reversed turn sequence Pm , . . . , P1 .
By optimal play we mean the outcome of a subgame perfect Nash equilibrium, which makes sense for either partially or totally ordered preferences.
(We will recall the definition below.) By the division of the food we mean the
final partition of morsels among the players, without regard to the order in
which they were chosen. We will prove a more general statement that also
applies to mixed tables of knights and louts (and even to individuals who act
as knights on some turns and louts on others), which we state as Theorem
4 below. Although boorish louts and gallant knights are very different, the
theorem gives a sense in which they are “dual” to each other.
Nash equilibria in general are hard to compute. In our setting, we have a
sequential game of perfect information which has a succinct description (i.e.,
the length of the game description is only logarithmic in the size of the game
tree). For such games, the problem of determining whether a given outcome is
an optimal play is Pspace-complete by a reduction to the quantified boolean
formula problem [1, Theorem 5.8]. The significance of Theorem 1 and Theorem
A DUALITY PRINCIPLE FOR SELECTION GAMES
3
4 is that they allow us to compute the food division resulting from optimal
play when some or all of the players are gallant knights – namely, reverse the
turn order and play as boorish louts. The food division when all players are
boorish louts is trivial to compute in linear time (in the size of the input, nk).
Theorem 1 and Theorem 4 are strong in the sense that our definition of gallant
knights makes only fairly weak assumptions (i.e., we assume only that their
preferences extend fairly weak partial orderings). It is somewhat surprising
that these weak assumptions are sufficient to determine the division of food
under any optimal play.
Related work. In a two player game where all morsels are ultimately consumed and both players are gallant knights, we can swap the roles of <1 and
<2 to obtain the game in which two players take turns selecting morsels each
with the aim of maximizing her own total enjoyment (but unlike boorish louts,
they do not discount the future). Kohler and Chandrasekaran [4] described
the optimal play for this latter game in 1971 (see also [3, Ch. 2] and [5] for a
recent survey).
Theorem 1 thus generalizes the 2-player theorem of [4] to a class of games
with any number of players. Curiously, we do not know of any efficiently
computable equilibrium for the most obvious generalization of the KohlerChandrasekaran game, in which k ≥ 3 players take turns selecting items and
each tries to maximize his own total enjoyment. Brams and Straffin point out
a number of pathologies that arise in this game [2].
Enjoyments and utilities. One concrete way to realize an extension of the
pairwise comparison ordering on plates is to assign enjoyments ei (a) which
are real numbers representing the value of morsel a to player i. We require
ei (a) 6= ei (b) for a 6= b, and we can say that one plate is more enjoyable
than another for player i if the sum of the enjoyments
is larger. Given a
P
play sequence a = (a1 , . . . , am ), we let Ei (a) = t : Pt =i ei (at ) denote the total
enjoyment that the ith player receives. Each leaf a of the game tree in Figure 1
is labeled by the enjoyment vector (E1 (a), E2 (a)).
We also remark that a concrete way to realize a preference ordering on
game outcomes for the ith player is to use a utility function that assigns a
real number to each play sequence a. An example of a gallant knight’s utility
function is
X
Ui (a) = αEi (a) +
Ej (a).
j6=i
for sufficiently small α > 0. Since Theorems 1 and 4 hold for any extensions of
the knight and lout orderings, in particular they apply when preferences are
defined by utilities in this way.
Equilibria for games with partially ordered preferences. The games we
consider will involve finitely many states and outcomes and can be represented
by decision trees such as Figure 1. A game outcome is a leaf of the tree. Each
player i has a partial ordering <i on game outcomes. A strategy for player i
is a way of assigning a decision to each game position in which player i has
the move. Each game position is represented by a node of the tree, and player
4
LEVINE, SHEFFIELD, STANGE
(2, 3)
a
b
(2, 3)
b
(1, 1)
c
(1, 1)
Player 1
c
(2, 3)
c
a
(2, 3)
b
(2, 3)
(0, 2)
c
(0, 2)
(1, 1)
c
a
(2, 3)
a
(2, 3)
(0, 2)
b
(0, 2)
Player 2
b
(1, 1)
a
Player 1
(1, 1)
Figure 1. Example of a game tree for 2 gallant knights and
3 morsels which has more than one optimal play. The morsels
a, b and c have enjoyment vectors (1, 0), (2, 1) and (0, 2) respectively. Each edge is labeled with the morsel eaten by the
moving player. Each decision node contains one arrow pointing
to a child. The collection of all arrows represents an equilibrium
pair of strategies, one for each player. Each vertex is labeled by
the enjoyment vector of the contingent outcome determined by
the equilibrium. The path of arrows from the root (left thickened path) is an optimal play. The pair of strategies would still
be an equilibrium if Player 1 chose the middle child instead of
the left child for the first move; the resulting optimal play is the
dotted path from the root. Note that the division of food is the
same in either case: Player 1 eats morsels a and b and Player 2
eats morsel c; only the order of eating changes.
i’s decision from a given position is represented by drawing an arrow from
the corresponding node to one of its children. A k-tuple of strategies (one
strategy for each player) determines a directed path from the root of the tree
to a leaf representing the outcome of the game. The arrows also determine,
for each node of the tree, a “contingent outcome,” which is the leaf reached
by following the arrows starting from that node.
Given a k-tuple of strategies s = (s1 , . . . , sk ), for each node x of the game
tree let Os (x) be the set of contingent outcomes corresponding to the children
of x. We say that s is a subgame perfect Nash equilibrium if for each node x of
the game tree, the contingent outcome of x is a maximal element of Os (x) with
respect to <i , where i is the player to move at x. A root-to-leaf trajectory
is called an optimal play if it arises from such an equilibrium. It is always
possible to construct such an equilibrium inductively (drawing decision arrows
one layer at a time starting from the bottom of the tree). If each player has a
total preference ordering on game outcomes, then this equilibrium is unique.
If the orderings <i on game outcomes are partial instead of total, then the
equilibrium may not be unique.
The following proposition is immediate from the definitions, and relates
equilibria for games with partially and totally ordered preferences on outcomes.
Proposition 2. Suppose that A and B are k-tuples of partial orderings on
game outcomes (one for each player), and each player’s ordering in B extends
A DUALITY PRINCIPLE FOR SELECTION GAMES
5
his ordering in A. Then any equilibrium for B is also an equilibrium for A.
Conversely every equilibrium for A is also an equilibrium for some k-tuple of
total orderings that extends A.
Theorem 1 states that the knight and lout partial orderings contain enough
information to determine how the food is divided under any optimal play
(though, as Figure 1 illustrates, they do not necessarily determine the order in
which selections are made). Another way to say this is that if we inductively
choose a subgame perfect Nash equilibrium with respect to the partial ordering
(i.e., at each node we draw an arrow to any one of the children corresponding
to an outcome that is maximal with respect to the deciding player’s partial
ordering) then the resulting optimal play determines a partition of the morsels
that does not depend on any of the choices we made along the way.
The following observation is also immediate. It says that edges of the decision tree that correspond to unambiguously “bad” decisions can be removed
without altering the set of optimal plays.
Proposition 3. Suppose that e1 , e2 , . . . is a finite sequence of edges of the
decision tree (corresponding to particular decisions made at nodes x1 , x2 , . . .)
and that there is no optimal play for any subgame beginning at any xj in which
a player uses the edge ej . Then removing e1 , e2 , . . . (and the subtrees beneath
them) from the decision tree does not alter the set of optimal plays.
Proof of the main result. We will work in slightly greater generality by allowing mixed nature individuals. A mixed nature individual can be understood
as a gallant knight who knows she will be required to act as a boorish lout on a
certain pre-determined subset of her turns called her lout turns (i.e., the edges
of the decision tree corresponding to other moves she would have made at that
step are just removed). From the outset of the game, both the lout turns of
each individual and the rankings <i on morsels are common knowledge to all
players.
Theorem 4. In any game of mixed nature individuals, all optimal plays lead
to the same division of food, which is the division obtained by pushing all knight
turns to the end, reversing the order of the knight turns and then converting
all knight turns into lout turns.
For example, a game with 3 players taking 6 moves in the order
1(knight), 2(knight), 2(lout), 3(lout), 3(knight), 1(lout)
would become
2(lout), 3(lout), 1(lout), 3(lout), 2(lout), 1(lout).
The original knight turns, in bold, have been pushed to the end, reversed,
and converted to lout turns. This procedure determines a bijection between
the turns of the original game and the turns of the resulting lout game. Corresponding to the unique optimal play of the lout game under this bijection is
a play of the original game, which the proof of Theorem 4 will show is in fact
optimal.
6
LEVINE, SHEFFIELD, STANGE
Lemma 5. Let G be a game of mixed nature individuals. Assume that each
player has preferences given by her knight ordering (rather than some extension
of that ordering) but is forced to behave as a lout on her lout turns. Let t be
the last turn on which any player acts as a gallant knight and suppose t < m.
Let G0 be the game obtained by reversing the turn order of Pt and Pt+1 . Then
a food division can be obtained by an optimal play a in G if and only if it can
be obtained by an optimal play a0 in G0 .
Proof. Consider the pair of turns t and t + 1 as a single turn that results in
a pair of morsels (a, b) being distributed, the first to the boorish lout and the
second to the gallant knight. Since the lout’s decision is deterministic, it is
the knight’s decision that determines the pair (a, b). If the knight goes second,
then the lout will always take her favourite morsel from those remaining on
turn t (call it a1 ) and the knight can choose any b from the remaining available
morsels. If the knight goes first, he can also choose any such b (in which case
the lout chooses a1 ). However, the knight has another option in this case: he
can choose a1 (which would cause the lout to choose a second favourite option,
a2 ).
If the knight and lout are actually the same player then the options (a1 , a2 )
and (a2 , a1 ) are equivalent, so this does not actually correspond to an additional choice. If the knight and lout correspond to different players, then the
choice (a2 , a1 ) is always less desirable to the knight than (a1 , a2 ), since it results in the lout getting a less desirable option during these two turns (and
since only louts remain, it does not affect the way food is distributed in subsequent turns). Thus, having the knight move before the lout adds a choice
for the knight (to each node of the decision tree corresponding to this combined move); but the added choice is one that the knight will never use under
optimal play in any subgame, so (by Proposition 3) adding or removing this
choice has no effect on the way food is divided under equilibrium strategies.
To phrase this differently, we have shown that a is optimal for G if and only
if a0 is optimal for G0 , where a0 is the sequence obtained from a by transposing
at and at+1 , unless Pt = Pt+1 and at is Pt ’s favourite remaining morsel, in
which case a0 = a. We remark that it was important in this argument that
all players had only the knight ordering on their preferences (rather than an
extension of that ordering), since is this is the reason that we could treat the
options (a1 , a2 ) and (a2 , a1 ) as equivalent.
Proof of Theorem 4. By Proposition 2, any equilibrium for players with extensions of the knight ordering is also an equilibrium for players with the knight
ordering. By Lemma 5, for any optimal play a of G, there is an optimal play
a0 of the game G0 obtained by pushing the last knight turn to the end, such
that a and a0 lead to the same division of the food. But when a gallant knight
moves last, she behaves the same way she would behave if she were a boorish
lout: she takes her favourite morsel remaining. Thus, a0 remains an optimal
play if we convert her into a lout. Repeating this procedure with each knight
in turn yields a game of only louts, which has a unique optimal play.
A DUALITY PRINCIPLE FOR SELECTION GAMES
7
Remark 6. The proof still works if the dinner table is a multiset, that is, it may
contain multiple identical copies of some morsels (five identical tempura yam
rolls, say). The only change to the argument is that, in the proof of Lemma 5,
one must consider the case that the lout has more than one identical favourite
item available at step t; but in this case, the lout will choose an item of that
type regardless of the way Pt+1 and Pt are ordered, so it does not matter which
of the two goes first.
Note, however, that Lemma 5 relies on the fact that when only lout turns
remain, the play is deterministic. For this reason, the proof does not apply
to the more general setting in which each player has only a partial preference
ordering on the morsels. If some player regards two morsels as incomparable
while another player has a preference between them, then on a lout turn, the
former’s choice may alter the outcome for the latter.
(2, 2, 3)
a
b
(3, 1, 3)
b
(3, 1, 3)
c
Players 2 & 3
c
(2, 2, 3)
c
a
(3, 3, 2)
b
(3, 1, 3) (3, 3, 2)
(2, 2, 3)
c
(1, 2, 2)
c
a
(2, 3, 1)
a
(2, 2, 3) (2, 3, 1)
(1, 2, 2)
b
Players 1 & 3
b
(1, 1, 1)
a
Players 1 & 2
(1, 2, 2) (1, 1, 1)
Figure 2. An example group decision game tree for 3 players
and 3 morsels. The enjoyment vectors for the morsels a, b, and
c are (0, 1, 2), (1, 2, 1) and (2, 0, 0) respectively. Each vertex
is labeled by the enjoyment vector of the contingent outcome
determined by the equilibrium. Each row is labeled with the
players to agree upon the move at that row. Each edge is labeled
with the corresponding morsel they share. Bold edges show the
optimal play b, a, c.
Another interpretation: games of group decision making. Consider
an alternate version of the dinner in which players take turns speaking. While
a player is busy speaking, the remaining k − 1 players jointly select one of
the morsels and share it among themselves. Play continues until all food has
been consumed (i.e., we assume the number m of turns equals the number
n of morsels). Different groups of k − 1 players may have different methods
of agreeing on a morsel to select as a group. Let us assume only that their
method is such that their choice is always Pareto efficient in the sense that
they will prefer play sequence a to b if a gives each member of the group
an equal or more enjoyable plate (and at least one member a more enjoyable
plate).
With these preferences, the game just described is equivalent (after reversing the enjoyment rankings for each player) to having the speaking player
8
LEVINE, SHEFFIELD, STANGE
be a gallant knight who decides on a morsel to eat for himself. To see the
equivalence, note that if player i partakes in a set of morsels Ai in the group
interpretation, that means he receives the complement Aci in the knight interpretation. The pairwise comparison ordering on sets Ai corresponds to the
comparison ordering on sets Aci when the morsel ranking is reversed, and the
Pareto ordering described above is exactly the knight ordering (here we use
that m = n, so that axiom (K2) is irrelevant).
The tree for a group decision dinner of three players is shown in Figure
2, along with the “enjoyment vector” labeling shown as in Figure 1. Let us
call an equilibrium (specified by a collection of strategies, one for each group)
conflict-free if no group can change a decision at a node unilaterally so that
any one member of the group gets a more enjoyable plate (even if another
member gets a less enjoyable plate). Theorem 1 shows the following.
Theorem 7. For the group decision dinner, every subgame-perfect equilibrium
is conflict-free, and all optimal plays result in the same division of the food.
The statement that every subgame-perfect equilibrium is conflict-free means
that as we construct an equilibrium inductively, at each node there exists at
least one choice that is optimal for all of the players in the corresponding
group. This is immediate from Theorem 1, which implies that if we are given
any equilibrium leading to some given outcome a, then all of the options at
a node x leading to outcomes that are incomparable to a (with respect to
the group’s Pareto ordering) must lead to the same division of food. If one
of these alternative outcomes was better for one player and worse for another
player, then it would have to lead to a different division of a food, contradicting
Theorem 1 (applied to the subgame beginning at x).
References
[1] C. Àlvarez, J. Gabarro and M. Serna, Equilibria problems on games: Complexity versus
succinctness, J. Comput. System Sci. 77:1172–1197, 2011.
[2] S. J. Brams and P. D. Straffin, Jr., Prisoners’ dilemma and professional sports drafts,
Amer. Math. Monthly 86(2):80–88, 1979.
[3] S. J. Brams and A. D. Taylor, The win-win solution, Norton, 1999. Nash equilibrium. In
Proceedings of the 38th annual ACM symposium on theory of computing (STOC), pages
71–78, 2006.
[4] D. A. Kohler and R. Chandrasekaran, A class of sequential games, Operations Research
19: 270–277, 1971.
[5] L. Levine and K. E. Stange, How to make the most of a shared meal: plan the last bite
first. Amer. Math. Monthly, to appear. arXiv:1104.0961
Department of Mathematics, Cornell University, Ithaca, NY 14853. http:
//www.math.cornell.edu/~levine
MIT Department of Mathematics, 77 Massachusetts Ave., Cambridge, MA
02139. http://math.mit.edu/~sheffield
E-mail address: [email protected]
Department of Mathematics, Stanford University, 450 Serra Mall, Building 380, Stanford, CA 94305 USA. http://math.stanford.edu/~stange
E-mail address: [email protected]